Historical Developments in Singular Perturbations

1. Front Matter

   Pages i-x

   PDF

2. Ludwig Prandtl’s Boundary Layer Theory

   • Robert E. O’Malley

   Pages 1-26

3. Asymptotic Approximations

   • Robert E. O’Malley

   Pages 27-51

4. The Method of Matched Asymptotic Expansions and Its Generalizations

   • Robert E. O’Malley

   Pages 53-121

5. Wendepunkts and Canards (Turning Points and Delayed Bifurcations)

   • Robert E. O’Malley

   Pages 123-140

6. Two-Timing, Geometric, and Multi-scale Methods

   • Robert E. O’Malley

   Pages 141-181

7. A Simple Multi-scale Procedure for Both Oscillatory and Boundary Layer Problems

   • Robert E. O’Malley

   Pages 183-213

8. Back Matter

   Pages 215-256

   PDF

This engaging text describes the development of singular perturbations, including its history, accumulating literature, and its current status. While the approach of the text is sophisticated, the literature is accessible to a broad audience. A particularly valuable bonus are the historical remarks. These remarks are found throughout the manuscript. They demonstrate the growth of mathematical thinking on this topic by engineers and mathematicians.

The book focuses on detailing how the various methods are to be applied. These are illustrated by a  number and variety of examples. Readers are expected to have a working knowledge of elementary ordinary differential equations, including some familiarity with power series techniques, and of some advanced calculus.

Dr. O'Malley  has written a number of books on singular perturbations.  This book has developed from many of his works in the field of perturbation theory.

• Asymptotic Methods
• Differential Equations
• History of 20th Century Differential Equations
• Multi-scale Methods
• Singular Perturbations
• ordinary differential equations

“This book covers a wide range of topics in the history of singularly perturbed differential equations. … This is a lucid book written in a deceptively easy style. The book will be useful both for specialists in the theory of asymptotical methods and for experts in various fields of applications. Moreover, it may equip students and young scientists with the essential tools needed for understanding the literature and doing research in the field of asymptotical analysis and its applications.” (Vladimir Sobolev, zbMATH 1336.34002, 2016)

"Such a book could be written only by an author who has devoted most of his working life to this topic, always with an eye on human interactions and the background of people in the field."

"The book is a great gift to the world of mathematical analysis."

Ferdinand Verhulst, Professor at the Mathematisch Instituut, University of Utrecht, the Netherlands, writing for SIAM May 15, 2015

• Department of Applied Mathematics, University of Washington, Seattle, USA

  Robert E. O'Malley

Robert O'Malley is a Professor Emeritus at the University of Washington. Dr. O'Malley has studied Electrical Engineering and Mathematics at the University of New Hampshire where he received his baccalaureate and masters degrees. He received his doctorate in Mathematics from Stanford University in 1966. Dr. O'Malley serves on a number of boards, including advisory committees for the National Institute of Standards and Technology, the Electric Power Research Institute, and Argonne National Laboratory.

Policies and ethics